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A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a constant... But I would like to know the more general meaning/definition of the term. Thanks.

Added: In particular, I would like to know what form such a R metric would look like e.g. $ds^2=adx^2+bdy^2$

P.S. I have tried searching for a definition, but I couldn't find a proper definition.

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If I ran across "locally Euclidean" in the context of a Riemannian metric, I would assume it meant that about (all but finitely many) points of the manifold, there is a neighborhood isometric to a neighborhood of $\mathbb{R}^n$ for some $n$. Example: square-tiled surfaces. But I'm not sure enough of this to make it an answer. –  Neal Apr 30 '12 at 16:48
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I've only ever seen locally Euclidean in the context of differential geometry, where it simply means that every point has a neighbourhood diffeomorphic to $\mathbb{R}^n$. I think it would be rather uninteresting if it had a neighbourhood isometric to a neighbourhood of $\mathbb{R}^n$. On the other hand, perhaps what is meant is that the manifold is infinitesimally Euclidean, in the sense of having a positive-definite inner product. (As opposed to infinitesimally Minkowski.) –  Zhen Lin Apr 30 '12 at 16:56
    
@ZhenLin: thanks, so would the most general form of the metric be $ds^2=(dx^2+dy^2)/f(x,y)$ for some function $f$? –  31415 Apr 30 '12 at 17:01
    
@31415: Yes, you can always choose coordinates so that the metric takes that form... locally. –  Zhen Lin Apr 30 '12 at 17:06
    
@ZhenLin: thanks again, if that is so, then the hyperbolic metric is also locally Euclidean, but the geodesics are not straight lines? Is there a term for a metric that is simply a scaled Euclidean metric for all the space on which it is defined so thattthe geodesics are straight lines? –  31415 Apr 30 '12 at 17:17

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Definitions can be found at: http://en.wikipedia.org/wiki/Locally_Euclidean.

I understand that Wikipedia may not be your favorite source, so if you can access googlebooks, searching for "locally euclidean" yields at least 6 out of the top 10 hits defined it as "each point has a neighborhood homeomorphic to $\mathbb{R}^n$." As previous commentors have also stated, sometimes the requirement for homeomorphisms is strengthened to require diffeomorphisms instead.

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